The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Claudio Procesi - One of the best experts on this subject based on the ideXlab platform.
-
the adjoint representation inside the exterior Algebra of a Simple Lie Algebra
Advances in Mathematics, 2015Co-Authors: Corrado De Concini, Paolo Papi, Claudio ProcesiAbstract:Abstract For a Simple complex Lie Algebra g we study the space of invariants A = ( ⋀ g ⁎ ⊗ g ⁎ ) g , which describes the isotypic component of type g in ⋀ g ⁎ , as a module over the Algebra of invariants ( ⋀ g ⁎ ) g . As main result we prove that A is a free module, of rank twice the rank of g , over the exterior Algebra generated by all primitive invariants in ( ⋀ g ⁎ ) g , with the exception of the one of highest degree.
-
on special covariants in the exterior Algebra of a Simple Lie Algebra
arXiv: Representation Theory, 2014Co-Authors: Corrado De Concini, Paolo Papi, Pierluigi Moseneder Frajria, Claudio ProcesiAbstract:We study the subspace of the exterior Algebra of a Simple complex Lie Algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie Algebra of traceless matrices, by the copies of the n-th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subAlgebra of the exterior Algebra generated by all primitive invariants except the one of highest degree.
-
the adjoint representation inside the exterior Algebra of a Simple Lie Algebra
arXiv: Representation Theory, 2013Co-Authors: Corrado De Concini, Paolo Papi, Claudio ProcesiAbstract:For a Simple complex Lie Algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the Algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior Algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.
Helmut Strade - One of the best experts on this subject based on the ideXlab platform.
-
Simple Lie Algebras of small characteristic VI. Completion of the classification
Journal of Algebra, 2008Co-Authors: Alexander Premet, Helmut StradeAbstract:Let L be a finite-dimensional Simple Lie Algebra over an Algebraically closed field of characteristic p>3. It is proved in this paper that if the p-envelope of adL in DerL contains a torus of maximal dimension whose centralizer in adL acts nontriangulably on L, then p=5 and L is isomorphic to one of the Melikian Algebras M(m,n). In conjunction with [A. Premet, H. Strade, Simple Lie Algebras of small characteristic V. The non-Melikian case, J. Algebra 314 (2007) 664–692, Theorem 1.2], this impLies that, up to isomorphism, any finite-dimensional Simple Lie Algebra over an Algebraically closed field of characteristic p>3 is either classical or a filtered Lie Algebra of Cartan type or a Melikian Algebra of characteristic 5. This result finally settles the classification problem for finite-dimensional Simple Lie Algebras over Algebraically closed fields of characteristic ≠2,3.
-
Simple Lie Algebras of small characteristic VI. Completion of the classification
arXiv: Representation Theory, 2007Co-Authors: Alexander Premet, Helmut StradeAbstract:Let L be a finite-dimensional Simple Lie Algebra over an Algebraically closed field of F characteristic p>3. We prove that if the p-envelope of L in the derivation Algebra of L contains nonstandard tori of maximal dimension, then p=5 and L is isomorphic to one of the Melikian Algebras. Together with our earLier results this impLies that any finite-dimensional Simple Lie Algebra over F is of classical, Cartan or Melikian type.
-
Simple Lie Algebras of small characteristic V. The non-Melikian case
Journal of Algebra, 2007Co-Authors: Alexander Premet, Helmut StradeAbstract:Abstract Let L be a finite-dimensional Simple Lie Algebra over an Algebraically closed field F of characteristic p > 3 . We prove in this paper that if for every torus T of maximal dimension in the p-envelope of adL in DerL the centralizer of T in adL acts triangulably on L, then L is either classical or of Cartan type. As a consequence we obtain that any finite-dimensional Simple Lie Algebra over an Algebraically closed field of characteristic p > 5 is either classical or of Cartan type. This settles the last remaining case of the generalized Kostrikin–Shafarevich conjecture (the case where p = 7 ).
Shrawan Kumar - One of the best experts on this subject based on the ideXlab platform.
-
hitchin s conjecture for simply laced Lie Algebras impLies that for any Simple Lie Algebra
Differential Geometry and Its Applications, 2014Co-Authors: Nathaniel Bushek, Shrawan KumarAbstract:Abstract Let g be any Simple Lie Algebra over C . Recall that there exists an embedding of sl 2 into g , called a principal TDS, passing through a principal nilpotent element of g and uniquely determined up to conjugation. Moreover, ∧ ( g ⁎ ) g is freely generated (in the super-graded sense) by primitive elements ω 1 , … , ω l , where l is the rank of g . N. Hitchin conjectured that for any primitive element ω ∈ ∧ d ( g ⁎ ) g , there exists an irreducible sl 2 -submodule V ω ⊂ g of dimension d such that ω is non-zero on the line ∧ d ( V ω ) . We prove that the validity of this conjecture for Simple simply-laced Lie Algebras impLies its validity for any Simple Lie Algebra. Let G be a connected, simply-connected, Simple, simply-laced Algebraic group and let σ be a diagram automorphism of G with fixed subgroup K . Then, we show that the restriction map R ( G ) → R ( K ) is surjective, where R denotes the representation ring over Z . As a corollary, we show that the restriction map in the singular cohomology H ⁎ ( G ) → H ⁎ ( K ) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case reLies on this cohomological surjectivity.
-
hitchin s conjecture for simply laced Lie Algebras impLies that for any Simple Lie Algebra
arXiv: Representation Theory, 2013Co-Authors: Nathaniel Bushek, Shrawan KumarAbstract:Let $\g$ be any Simple Lie Algebra over $\mathbb{C}$. Recall that there exists an embedding of $\mathfrak{sl}_2$ into $\g$, called a principal TDS, passing through a principal nilpotent element of $\g$ and uniquely determined up to conjugation. Moreover, $\wedge (\g^*)^\g$ is freely generated (in the super-graded sense) by primitive elements $\omega_1, \dots, \omega_\ell$, where $\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive element $\omega \in \wedge^d (\g^*)^\g$, there exists an irreducible $\mathfrak{sl}_2$-submodule $V_\omega \subset \g$ of dimension $d$ such that $\omega$ is non-zero on the line $\wedge^d (V_\omega)$. We prove that the validity of this conjecture for Simple simply-laced Lie Algebras impLies its validity for any Simple Lie Algebra. Let G be a connected, simply-connected, Simple, simply-laced Algebraic group and let $\sigma$ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G) \to R(K) is surjective, where R denotes the representation ring over $\mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology H^*(G)\to H^*(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case reLies on this cohomological surjectivity.
Bertram Kostant - One of the best experts on this subject based on the ideXlab platform.
-
on the Algebraic set of singular elements in a complex Simple Lie Algebra
arXiv: Representation Theory, 2010Co-Authors: Bertram Kostant, Nolan R WallachAbstract:Let $G$ be a complex Simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed Algebraic cone of singular elements in $\g$. Let ${\cal L}\s S(\g)$ be the (graded) ideal defining $\hbox{\rm Sing}\,\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\g$. Then ${\cal L}^k = 0$ for any $k
-
powers of the euler product and commutative subAlgebras of a complex Simple Lie Algebra
Inventiones Mathematicae, 2004Co-Authors: Bertram KostantAbstract:If \(\mathfrak{g}\) is a complex Simple Lie Algebra, and k does not exceed the dual Coxeter number of \(\mathfrak{g}\), then the absolute value of the kth coefficient of the \(\dim\mathfrak{g}\) power of the Euler product may be given by the dimension of a subspace of \(\wedge^k\mathfrak{g}\) defined by all abelian subAlgebras of \(\mathfrak{g}\) of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson’s 2rank theorem on the number of abelian ideals in a Borel subAlgebra of \(\mathfrak{g}\), an element of type ρ and my heat kernel formulation of Macdonald’s η-function theorem, a set Dalcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when \(\mathfrak{g}= \text{Lie}\,\mathit{Sl}(m,\mathbb{C})\)), and the homology and cohomology of the nil radical of the standard maximal parabolic subAlgebra of the affine Kac–Moody Lie Algebra.
-
powers of the euler product and commutative subAlgebras of a complex Simple Lie Algebra
arXiv: Group Theory, 2003Co-Authors: Bertram KostantAbstract:If $\frak g$ is a complex Simple Lie Algebra, and $k$ does not exceed the dual Coxeter number of $\frak g$, then the k$^{th}$ coefficient of the $dim \frak g$ power of the Euler product may be given by the dimension of a subspace of $\wedge^k\frak g$ defined by all abelian subAlgebras of $\frak g$ of dimension $k$. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson's $2^{rank}$ theorem on the number of abelian ideals in a Borel subAlgebra of $\frak g$, an element of type $\rho$ and my heat kernel formulation of Macdonald's $\eta$-function theorem, a set $D_{alcove}$ of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null $m$-core when $\frak g= Lie Sl(m,\Bbb C)$), and the homology and cohomology of the nil radical of the standard maximal parabolic subAlgebra of the affine Kac-Moody Lie Algebra.
Marco Pedroni - One of the best experts on this subject based on the ideXlab platform.
-
drinfeld sokolov reduction on a Simple Lie Algebra from the bihamiltonian point of view
Letters in Mathematical Physics, 1992Co-Authors: Paolo Casati, Marco PedroniAbstract:We show that the Drinfeld-Sokolov reduction can be framed in the general theory of bihamiltonian manifolds, with the help of a specialized version of a reduction theorem for Poisson manifolds by Marsden and Ratiu.
